### M Theory Lesson 120

A non-blacklisted professional physicist from Riverside, Ernest Ma, has a 2006 paper on the arxiv about neutrino mixing matrices. He discusses a derivation of tribimaximal mixing from the groups $S_3$ and $A_4$, the even permutations on four objects.

This relies on describing the mass eigenvalues using parameters $a$, $b$ and $d$ for which the masses become $a - b + d$, $a + 2b$ and $-a + b + d$. Setting $d = 0$ looks wrong, because the first and third masses would be equal and of opposite sign, but note the similarity between $a - b$ and $a + 2b$ and the expressions $1 - r$ and $1 + 2r$ which appear in the circulant matrices. In fact, in the $d = 0$ case Ma's mass matrix reduces to a 2-circulant in $a$, $b$ and $b$.

This relies on describing the mass eigenvalues using parameters $a$, $b$ and $d$ for which the masses become $a - b + d$, $a + 2b$ and $-a + b + d$. Setting $d = 0$ looks wrong, because the first and third masses would be equal and of opposite sign, but note the similarity between $a - b$ and $a + 2b$ and the expressions $1 - r$ and $1 + 2r$ which appear in the circulant matrices. In fact, in the $d = 0$ case Ma's mass matrix reduces to a 2-circulant in $a$, $b$ and $b$.

## 2 Comments:

A couple of comments about TGD view about CKM mixing.

In the case of quarks (and also leptons) rational algebraic physics approach gives powerful conditions on CKM matrix.

1. In TGD framework CKM matrix V=U^*D, reflects different topological mixing of U and D type quarks (quark generations correspond to genus g=0,1,2 for partonic 2-surface). The interpretation in terms of topological mixing is the new physical input.

2. p-Adic length scale hypothesis allows to separate the mass scale completely out of calculation so that one can work with integer valued conformal weights. From TGD point view the attempts to understand neutrino mixing algebraically without taking into account varying mass scale would work only if all neutrinos have the same p-adic mass scale.

3. The mixing of genera implies that the calculable mass squared contributions -or equivalently contributions to conformal weights - from modular degrees of freedom for g=0,1,2 mix too. Powerful number theoretical conditions are obtained since the modular conformal weights must remain integers in topological mixing.

4. I end up with a variational principle stating that an entropy assignable to the mixing is maximal so that kind of number theoretic thermal equilibrium state is in question. Fits naturally with p-adic thermodynamics.

The condition that U and D matrices are rational plus some rough inputs from Cabibbo mixing and CP breaking lead even without the variational principle to essentially unique CKM matrix.

Great, Matti! This summary clarifies a couple of things for me.

...quark generations correspond to genus g=0,1,2 for partonic 2-surfaceI'm still wondering what to do with those pieces of genus 0, 1 and 2 moduli in the twistor operad setting. We might need to sort out the triality logic axioms (higher topos/logos) before it becomes clear how such spaces model the axioms.

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